Bob,

It seems like if there's no one behind you and therefore no risk of hitting another car, you'd be better off letting go of the brake entirely: that way, the bus's kinetic energy that gets transferred to your car will simply push you along as if you were coasting in reverse. That never hurt you, did it? Also, the ease with which your car is pushed backwards will reduce the damage to your front bumper.

What causes the pain of whiplash or seatbelts boring into our chests or our unbuckled bodies flying through the windshield is the transfer of energy from the vehicle that crashes into us combined with our vehicle's resistance to move forward with that energy. If we had something equally as strong as brakes holding our bodies stationary, it seems like we'd be okay because we wouldn't move much and wouldn't feel the seatbelt or our head slamming into the back of the seat or anything like that. But we aren't fastened in that tight, so our bodies don't resist the force as strongly as the car does, so we lunge forward and then quickly backward (or, in the head-on collision scenario, backwards and then quickly forwards, I imagine). The pain is caused by the difference between what our bodies do and what the car does. The way to reduce the pain is to allow our car to move with us.

What happens when you accelerate real fast on your own? You get pressed back into your seat. What happens when a car crashing into yours from behind tries to "accelerate" your car but your brakes keep you stuck in one place? That's when your body lunges forward and then back against the seat. The force (acceleration) applied to us by the errant vehicle per se is not what causes pain or danger; it is that acceleration combined with our brakes and our seatbelts and our seat-backs.

So, obviously if there's someone behind you (or in front, in other scenarios), you apply the brake as hard as you can to save on insurance money and save the other driver from a big hassle; but if there's no one around, take the kinetic energy and let it propel you forwards or backwards and let your car gradually slow down, at the same rate that your own body does.
 

P.S.

But, it should be added that if that bus is turning left and cutting it awfully close to your front bumper, you don't want to let go of the brakes and inch forward to the point that the bus does hit your car where it would have missed it by inches otherwise!
 

John,

I think you are totally wrong. :) However, your (alleged) mistake is quite natural, and is why I asked the question in the first place.

The way the bus transfers its energy to you is through your car (seat belt, friction of your butt on the seat, steering wheel perhaps, etc.). The bus doesn't physically touch you. So the harder you have on the brakes, the more the bus slows down when it smacks your car, and the less your car needs to accelerate your body.

Think of it this way: I am at a dead stop at the light. You are saying I will experience less force if I end up coasting backwards than if I move backwards at a much lower speed. Do you see why that makes no sense? How do I end up "coasting" backwards at a faster speed, except through a larger force being applied?

Or, suppose I were in a tank that barely moved when the bus bounced off the front of it. Are you claiming I would be hurt more because of that?
 

Sorry for another long response when you have blogs to update, congressmen to pwn, economies to save, and enough other bloggers to obsess over regarding cap-and-trade policies, but this is really interesting and I don't get to think about math or physics much anymore.

I think I was assuming something you weren't, which I probably shouldn't have assumed given the situation you described. I was assuming the bus would hit you head-on fast enough to jar your car backwards pretty strongly, after which you would stop suddenly with the brakes applied but stop gradually with the brakes off; but if the bus is only in the middle of a turn, you're right: it probably wouldn't move your car back at all and your brakes would keep your car perfectly still, as if the bus were hitting an immovable tank. (In that case, the nudge the bus would give you with the brakes off certainly wouldn't hurt any adult, either; and the decreased resistance your bumper gives the bus might result in less damage to your bumper than if you stayed there as solid as a rock.)

So first let's picture our car going in the forward direction, which we're all more familiar with. If we accelerate quickly going forwards, we lean back into our seat more forcefully. No big deal. If we decelerate quickly in the forward direction, we lunge forward into our seat belt, which, in all modern cars, snaps taut and doesn't give any when the car stops really fast (or maybe it's when the brakes are applied strongly...not sure). The problem is that after we lean forward into the seat belt, we snap backwards into our seat-back and this can lead to whiplash.

Switching things around, if we accelerate quickly in the reverse direction, we lunge forward into the seat belt. If we decelerate quickly in the reverse direction, we're pressed back into our seat.

The way I see our contrasting positions is this: I say with the brakes off, the resultant higher rate of backwards-acceleration (and therefore more forceful and painful lunging into the seat belt) is not so bad because you won't whip backwards very strongly into the seat-back because you'll coast to a stop instead of decelerating very quickly to a stop. You say it's better with the brakes on because you won't accelerate backwards as fast and so won't lunge forward as much, and even though the rate of deceleration due to the brakes is much higher, it won't be so bad because you weren't going as fast to begin with. I say with the brakes applied, even starting from a lower speed, the nearly-immediate deceleration will whip you back pretty fast. You say with the brakes off, the higher initial acceleration will hurt you lunging forward and/or whipping back more, even though the backwards-deceleration admittedly is at a lower rate.

As for which will hurt more, the seat belt or the seat back, I don't know...

Here's something else to help you and I think about this acceleration-deceleration issue. When I was learning how to drive, I didn't stop at stop signs very smoothly. I would try to slow down gradually, but then all of a sudden the stop sign was there so I had to press the brake really hard. And my mom or dad would lean forward into the seat belt the way we all do when we stop too suddenly. This is after going like 4 or 5 miles an hour in the last few feet leading up to the stop sign. But a sudden, high rate of deceleration, such as after a fast head-on collision with our brakes applied strongly, makes us lunge way forward into our seat belts and then whip back into our seat backs.

Do you think accelerating backwards really fast (i.e., with no brakes) and then coasting to a stop would still make you whip back into your seat? Maybe, but I don't know and I don't want to find out empirically. I think laying down on the brakes would make it more forceful, even given the slow starting speed, as demonstrated by the meager speeds required to make my parents lunge forward at the stop sign. With the brakes off, you'd be starting from a higher speed, but the increased time would make the deceleration a much lower number. With the brakes on, even with a slower starting speed, the greatly decreased time makes the deceleration value higher (I postulate). In acceleration/deceleration, the time (denominator) is squared, remember, meaning time has the most influence on the value...

So maybe given the low speed and the ease with which we can make our passengers lunge forward, you'd want to reduce the backwards-acceleration (forward-lunging) from the bus collision by as much as possible, because obviously a little bit goes a long way; or maybe, on the other hand, you'd want to reduce the later deceleration by as much as possible because that's when you'll snap back into your seat and obviously a little bit of deceleration goes a long way.

As for the immovable tank example, the acceleration and deceleration are both zero, so that is why you wouldn't feel anything. I don't think that really solves anything.

Either way, it's well-known that you're a materialistic, fat-cat capitalist economist, so you're only concerned with money and material things, so you'd want to reduce the damage to your Aston Martin or Hummer by as much as possible, which means you should let your car coast backwards so your front bumper and hood resist the bus's force as little as possible.